Critical behavior in annealed and unannealed crystals of ...Benzil (C6HsCOh is a molecular crystal which has a crystallographic phase transition at -83.5 K. Since the discovery of

Critical behavior in annealed and unannealed crystals of benzilA. Yoshihara, E. R. Bernstein, and J. C. Raich

Citation: The Journal of Chemical Physics 77, 2768 (1982); doi: 10.1063/1.444191View online: http://dx.doi.org/10.1063/1.444191View Table of Contents: http://aip.scitation.org/toc/jcp/77/6Published by the American Institute of Physics

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Critical behavior in annealed and unannealed crystals of benzila)

A. Yoshihara, E. R. Bernstein, and J. C. Raich

Department of Chemistry and Physics, Colorado State University, Fort Collins, Colorado 80523 (Received 21 December 1981; accepted 1 June 1982)

Simultaneous Brillouin-correlation light scattering results obtained under an external applied stress pulse are presented for annealed and unannealed (as grown) crystals of benzil. Critical fluctuations are characterized for the LA a -axis mode governed by the elastic constant c II and are shown to contribute to this elastic anomaly in the high temperature phase. Both a central peak intensity anomaly and a critical relaxation time anomaly are characterized for unannealed crystals. These effects disappear in well-annealed crystals and without the external stress pulse even in as grown crystals. The critical relaxation behavior in benzil is coupled to the external stress compression wave generated by the mechanical refrigerator used to cool the sample. The latter interaction and relaxation anomaly are observed in polarized scattered light with the stress applied along the b (or a) axis as required for a c II coupling. These observations are qualitively analyzed employing the theory of anelastic solids with higher order fluctuation terms incorporated into the free energy expansion. The relationship of this description to the Halperin-Varma theory of defect induced central peaks is outlined.

I. INTRODUCTION

Benzil (C6HsCOh is a molecular crystal which has a crystallographic phase transition at - 83.5 K. Since the discovery of this phase transition through birefrin-gence studies, 1 many experimental techniques have been employed to investigate this system. 2- 8 The high tem-perature phase has been determined to possess a trigonal structure Dj(P3 i 21) with three molecules/unit cell. The low temperature phase structure is aSSigned as C2.

The high symmetry phase of benzil is piezoelectric and the phase transition can be considered a ferroelec-tric-ferroelastic transition although the dielectric anomaly is quite weak. However, a symmetry argu-ment based on molecular arrangement suggests that the spontaneous polarization due to the permanent di-pole moment of the benzil molecule cannot be reversed by an external electric field. Benzil can thus be con-sidered as belonging to the class of pyroelectric crys-tals.

Raman scatteringS and infrared spectroscopy6 have characterized a soft E mode which splits in the low symmetry phase. Moreover, Brillouin studies7• 8 re-veal soft transverse acoustic (T A) phonons governed by an elastic instability in C44 and a soft longitudinal acous-tic (LA) phonon governed by an elastic instability in cn' A Landau mean field theory fully accounts for the T A soft phonon behavior but cannot account for the Cll anomaly. 8

Actually, the Cu elastic constant can be decomposed into two independent contributions, ~ (cn + C12) and CS6 as follows:

Cll = t(cn + C12) + t(cu - C12) = t(Cll + C12) + C6S

C66 evidences some softening, but it is too weak to ac-count for the entire cll anomaly. The {(Cll + c 12) elastic constant is defined as the coefficient of (ei + e2)2 in the elastic energy expression for D3 symmetry; the strain (ei + e2) is one of the elastic basis functions transforming

a) Supported in part by a grant from AFOSR.

according to the irreducible representation A of the group Ds. Group theory predicts that any bilinear cou-pling between this basis and the E-symmetry order parameter is forbidden; the allowed coupling in lowest order is an electrostrictive-type interaction A® [E2 J in which the [ J Signifies the symmetric square. The ex-plicit expression of this interaction is given by8:

(el + e2) (Q~ + Q~)

in which Q1 and Q2 represent components of the doubly degenerate E-symmetry order parameter" Through this coupling, hCll + C12) possesses a critical anomaly which is stronger than the C66 anomaly and accounts for the cn observed behavior. The above interaction also introduces a step discontinuity in the elastic con-stant at the transition temperature. The observed discontinuity substantiates a strong anharmonic fluctua-tion dominated interaction (See Fig. 2 of Ref. 8). Brillouin scattering experiments have consequently re-vealed a clear deviation from mean field Landau be-havior in this molecular crystal.

An x-ray diffraction experiment3 in the low tempera-ture phase at -74 K suggests a broken translational sym-metry. Toledan09 has developed a theory for the phase transition based on the observed broken translation sym-metry in the (0001) plane of the hexagonal cell. Toledano introduced a three-dimensional Brillouin zone boundary order parameter (at the M point) which is assumed to be driven by the zone center instability through anhar-monic interactions.

Reference 8 discusses the observed elastic anomalies and the zone center instability employing a Landau mean field theory developed for a ferroelastic phase transition in D3d symmetry. It also treats qualitatively a contribu-tion to the C u anomaly due to the zone boundary order parameter; it is demonstrated in Ref. 8 that both zone boundary and zone center contributions to the Cll be-havior have the same form for the anharmonic coupling interactions considered. Unfortunately, elastic mea-surements mentioned above cannot distinguish which contribution is more important.

2768 J. Chern. Phys. 17(6), 15 Sept. 1982 0021-9606/82/182768·11$02.10 © 1982 American Institute of Physics

Yoshihara, Bernstein, and Raich: Critical behavior in benzil 2769

In the past ten years, a number of studies have dem-onstrated that, in systems with a soft phonon response function, central peaks [large increases in Rayleigh intensity as (T - Tc) - 0] can be observed. The initial report of this phenomenon was for a neutron scattering study of SrTiOs. 10 Since then, many experimental ob-servations and theoretical models have appeared dealing with this manifestation of critical behavior. 11 Phenome-nologically, the central peak can be understood as a consequence of bilinear coupling between soft mode(s) and a relaxator. Heat diffusion modes, microdomain motion, and defect motion have been considered as pos-sible candidates for the relaxator.

Brillouin scattering is one of the most powerful techniques for the study of critical phenomena. The first report of a central peak observed by Brillouin scattering is due to Lagakos and Cummins12 on the fer-roelectric phase transition in KH2P04(KDP). Since this discovery, the most extensive investigations on the origin of this phenomenon in KDP have been carried out by the Cummins et al. 13,14 Durvasula and Gammon, 15 Courtens, 16 and Courtens and Gammon. 17 Durvasula and Gammon argued that the central peak is VH polarized and static in origin as "speckle" interference fringes are observed around the transition point. These authors suggested that "static" defects are responsible for the central peak observations in KDP and suggested that natural abundance deuterium substituted for hydrogen in the crystal was the important defect. Courtens16(a) developed a mean field theory based on this pOint of view and was able to explain quantitatively the relative strength and temperature dependence of the observed central peak phenomena; these ideas and results, how-ever, are not substantiated by further experiments. 16(b) Courtens16(b) also confirmed that the KDP central peak can be strongly suppressed by annealing. For annealed crystals (140°C, 18 h), only a uniform laser light col-umn could be observed in VH or polarized light even at Tc + 0.3 K. This is to be contrasted with the static de-fect induced speckle interference pattern observed in as grown samples.

Mermelstein and Cummins13 observed a new central peak in KDP just below the transition temperature. The new central peak is considered to be due to coupling between the ferroelectric soft mode and the thermal diffusion mode through the temperature dependence of the order parameter. Unlike the aforementioned static central peak, this phenomenon is considered to be a dynamiC central peak. The dynamiC central peak is forbidden in the high symmetry (high temperature) phase because the soft mode and thermal diffUSion mode pos-sess different symmetry. Applying an external electric field to KDP and thereby reducing the high temperature phase symmetry, Courtens16(c) found that the specific heat and thermal diffusivity extrapolated to zero field are temperature independent immediately above the transition.

Similar phenomena have been observed in KH3(SeO ) 18 3 2 (KTS) by Yagi et al. The central peak in this crystal is VV polarized and strongly depends on the history of the sample. Even though KTS has a soft mode which

can be observed in VH polarization, the polarization for the central peak suggests that there is no direct coupling between the "central mode" and the soft mode. On the other hand, similar soft T A mode and central peak behavior have been observed for the deuterated salt in VH polarization. This central peak is indepen-dent of the history of the sample. Moreover, highly deuterated samples evidence less divergent behavior. The origin of the DKTS central peak phenomena is considered to be the presence of impurities (protons) which can couple to the soft x-y shear mode. 19 Although the origin for the KTS behavior is not yet apparent, in-homogeneous strains introduced in the process of crys-tal growth and shaping have been suggested as a possi-bility .

Lead germanate Pb5Ge30 11 is another well-studied crystal which exhibits both static and dynamic central peak phenomena in light scattering spectra. 20 Lyons and Fleury assigned the dynamic central peak in this system, which shows a log (Tc- T) divergence, as re-lated to the soft mode phonon density fluctuations inter-action proposed by Cowley and Coombs. 21 They further aSSigned a static central peak which displays a power law temperature dependence with index 0.85 ± 0.2 as due to static symmetry breaking defects. The width of the dynamic central peak is as low as 4 GHz near Tc and the width of the narrow (static) central peak is less than 2 MHz and unresolved by Brillouin techniques.

In addition to studies by Fabry-Perot Brillouin scat-iEring techniques, more direct approaches to the reso-lution of the narrow width of the central peak have been pursued for KDP, 22 quartz, 23 and Pb5Ge30u. 24 Un-fortunately, no correlation function could be observed in a range of 1Hz-50 MHz. To the best of our knowledge, successful observation of the central mode (dynamic or statiC) by photon correlation spectroscopy has not as yet been reported. Using small angle scattering, it is possible to cover the time domain in which dynam-ical central modes are to be expected, at least around the transition temperature. At present, it is not un-derstood why central peaks have not been observed by correlation spectroscopy under these conditions.

A theory of the defect induced central peak has been developed by Halperin and Varma. 25 They classify de-fects by the symmetry of the occupied site and the dy-namics of the entire cell and calculate structure fac-tors in two cases. For the "relaxing defect cell, " theory predicts a narrow central peak with a width given by a renormalized relaxation time and a pair of soft phonon features. The defect cell stabilizes the low temperature structure due to a tendency of all the de-fects to order in such a manner as to favor the same orientation of the order parameter. The actual defect generated transition temperature is higher than the perfect crystal transition temperature. For the "frozen impurity" model, the defect or impurity merely fluctu-ates about its pOSition and the obtained transition tem-perature becomes lower than the perfect crystal one. The dynamical structure factor in this case consists of three components: a zero width central peak for which the intenSity diverges as T- Tc; a finite width

J. Chem. Phys., Vol. 77, No.6, 15 September 1982

2770 Yoshihara, Bernstein, and Raich: Critical behavior in benzil

central peak for which the width decreases as T- Tc; and a pair of phonon peaks for which the frequency de-creases as T- Tc initially, but then begins to increase before Tc is reached. The experimental results already discussed seem to be consistent with a general relaxing defect model.

Recently, a neutron inelastic scattering experiment revealed a central peak response in addition to a soft mode rotational response near a structural phase tran-sition in another molecular system, chloranil. 26

During the course of our Brillouin investigations on benzil, the Rayleigh intenSity, as observed in the Bril-louin spectra along the a axis, was found to show a rapid enhancement just above the transition temperature [(T- Tc)/Tc$2 X 10-

2]. A similar, but less pronounced,

central peak associated with a soft T A-mode response was observed in the ferroelastic phase tranSition in sym-triazine by Brillouin scattering. 27 An attempt was also made to detect a relaxation mode by correlation spectroscopy covering the time domain between 1 and 10-6 s. Unfortunately, no relaxation process could be detected within this range for the triazine transition.

In this report, we will present more extended inves-tigations of the central peak in benzil by Brillouin scat-tering and correlation spectroscopy achieved under an external applied stress pulse. It is found that observa-tion of the central peak response in benzil is strongly coupled to not only crystal quality, but also the external pulse. In the presence of the external stress pulse, "as grown" samples show a central Rayleigh intenSity peak at Te , while well annealed crystals do not. Cor-relation spectroscopy reveals that, in as grown but not in annealed crystalS, a critical (slow) relaxation is excited if a periodic external direction dependent stress is applied to the crystal. This relaxation or correla-tion time can be modeled by an anisotropic structural defect in the crystal employing the theory of anelastic solids. It is concluded that in order for the critical dynamiCS of benzil to be manifested in the Rayleigh central response light scattering spectrum, amplifica-tion through a local stress/dielectric tensor inhomoge-neity is apparently necessary. Similar results have also been observed for chloranil and will soon be re-ported in a separate publication. Negative results of such experiments are also discussed for NH 4Cl and triazine. Urea, which does not undergo a phase transi-tion between 300 and 15 K, evidences an external stress related correlation function but without any anomalous temperature behavior.

II. EXPERIMENTAL

The experimental apparatus for the BrillOuin scatter-ing and correlation spectroscopy is such that both mea-surements can be made Simultaneously by splitting the scattered light. Figure 1 shows an optical diagram of this apparatus, including the data acquisition systems. Details of the Brillouin scattering system alone have been already reported. 8 For the simultaneous mea-surement of Brillouin and correlation spectra, a beam splitter is placed behind a pinhole and the resultant beams are introduced into each arm of the Brillouin/

----S±~-+ --1 .NOLE..". - I Ar -LASER

-1- P.H. L P B 5·'-.:K I 't-. B S

L I \ I, 1 L ~~ W

i M \--7 M RAMP FPI _,_ PH. GENERATOR I---�or_r-w

STABILIZER

MONIIDR

L

PH. i

-I-PH \

~~ MC'AIIIIDR

FIG.!. Optical diagram and data acquisition system for the simultaneous measurement of Brillouin scattering and corre-lation spectroscopy. Notation: L = lens, P. H. = pinhole, M = mirror, P=prism, B.S. = beam splitter, P.M. =photomulti-plier tube, and P.A. D. =preamplified/discriminator.

correlation spectrometer. It has been found that such a measurement procedure is necessary to obtain re-producible results and to obtain proper correlation in-tensity 0

Apparatus for correlation spectroscopy is quite sim-ple and consists of a pair of pinholes (500 JJ.m diameter each) spaced 20 cm apart and followed by a thermo-electrically cooled photomultiplier tube (RCA C31034A-02). A lens is placed just behind the beam splitter to increase Signal and a narrow band filter for 5145 A is placed in front of the pinholes to attenuate Raman scat-tered and fluorescence light from the sample. Pulses from the phototube are amplified and discriminated and then fed into a real time autocorrelator (Malvern K7025). The 64 channel correlator has a range of sampling times from 10-7 to 10-1 s, effectively covering a time domain between 10-6 and 1 s. The correlator has two modes of operation for autocorrelation: a zero clipped correla-tion mode and a full correlation mode. In this benzil study, the full correlation function mode was employed.

A desk top computer (HP 9845S) controls both the correlator for Rayleigh scattering and the multichannel analyzer for Brillouin scattering experiments. The computer accepts a data set from the correlator at the end of a preset experimental detection time and calcu-lates the intenSity autocorrelation function by a least squares fitting technique employing the simple form

g(2)(t) = 1 + 0 +Ae-2t IT ,

in which A, 0, and T are the fitting parameters. 28 A finite value of 0 can be due to real scattering processes

J. Chern. Phys., Vol. 77, No.6, 15 September 1982


. e. .. ..... e. • •••

........ _ '-... .. • ..J'-..

8 1· ..... ·1 I ........ j I ~-LJ-~~--~~--~ --'0 10 20 30 40 50 60

QiANNa

FIG. 2. Intensity correlation function for benzil at very low frequency showing the 3. 3 Hz modulation applied to the sample through the refrigerator. Sampling time is 10-2 s and the cor-relation function is repeated each 30 channels. Thus, one pe-riod corresponds to 10-2 x30=0.3 s. The parasitic shock wave which causes the stress pulse to the sample occurs on a much faster time scale as described in the text.

with larger relaxation times relative to the one being studied, to misnormalizations caused by fluctuations in laser intensity, or to other external sources. We will discuss this point more fully below.

The coefficient A has a value between 0 and 1 de-pending on sampling time Ts , dead time Td , and pinhole size. For the full correlation function mode of opera-tion, the A factor can be expressed by a product of three independent contributions as foUows29:

A=At(Te/ Ts)A2(Ts/Td)A3(ScoJS) ,

in which Te is the coherence (relaxation) time, Scob is the coherence scattering area (SCOh=A2/dn), and S is the effective detector area. For larger values of T e/ Ts and Tj Td, At and A2 approach unity. On the other hand, As approaches unity for ScoJS> 1 and approaches ScoJS for ScoJS« 1.

The actual experimental conditions are

Te/Ts -60, Ts/Td-l0, and ScoJS-l/7 .

In an experiment, the A factor will be determined mainly by A,( ScoJ S) and is approximated by

S A"'~-0.15 .

Changing the 500 /.Lm pinholes to 200 /.Lm pinholes causes the A valve to increase, but the Signal level to decrease; the 500 /.Lm pinhole set was used throughout the experi-ment. Estimated error (standard deviation) in the cal-culated relaxation time is about ± 10%.

Single crystals of benzil are grown from xylene satu-rated solutions by slow evaporation. Two different types of 90° scattering samples were prepared: "as grown" from solution and annealed at 50°C for -48 h.

A closed cycle He refrigerator (CTI 21) is used to

obtain temperatures as low as 15 K. The sample is affixed to a copper block using G. E. varnish 7031 and then placed in a copper scattering cell wrapped with a heater. Temperature is controlled with a proportional controller through a sensor diode placed on the copper cell. A copper-constantan thermocouple is placed just below the sample and affixed to it by varnish in order to measure temperature at the sample more precisely. Temperature fluctuation during the data accumulation time of about one hour is no more than ± 1 /.L V or about 0.05 K around 80 K,

Each refrigerator cycle is roughly 0.3 s long (3.3 Hz engine). A correlation function due to this engine cycle is represented in Fig. 2. For shorter sampling time measurements, this refrigerator correlation function is superimposed on the sample related correlation func-tion and gives a finite value to the parameter


3r-

,,~ 2r-'Q

f...1 ., ___ •• ___ .'I •• .;_!!~~~-"....!-•• - '-T~-J--·-!.

Or-

.)(

080" 0 ----"- -41 o .0 0 .......

:.:.: 00 -2j •• • 0

-~-- ........ -----~-.-. ..... -.·--~-o-~ I

gsO 00 85 90 TEMPERATURE (K )

FIG. 4. Temperature dependence of the relaxation time and counting rate/sampling time for an as grown crystal. Broken lines show base lines for the relaxation time (Te) and counting rate assuming that they would not change at T c' Full circles for the counting rate results were calculated from the anom-alous part of the relaxation time using Eqs. (14) and (22). The calculated value at 84 K shown by ® was used to normalize the result. The relaxation time and the intensity anomaly are thus strongly correlated with one another as indicated in the text [see Eqs. (22) and (23)). structure in both plots is reproduc-ible over various samples and for a single sample at many positions across a crystal face.

g


qll(OOI}

I

80 90 100 TEMPERATURE, K

ql/UOO} +

110

::i


1 2 2 1 S 2 1 2 2 2 1 f.Crl + Cr2\ 2 1 0 2 1 0 2 2 1 0 Z 2 F=Fo+"2 a(QI +Q2)+sb(Ql- 3Q1Q2)+"4 C(QI +Q2) +2 \ 2 } (el +e2) +"2Csses+"2 C44(e4+ e5) +"2 c66(e6+ e7)

+ crs (el + e2)eS + Cr4(e4e7 - e5e6) + A(e4Q1 - esQ2) + B( e7Ql + esQ2) + C( el + e2) (Q~ + Q~) + • • • (1)

in which a = ao( T - To), e7 = el - e2, and cg6 = (C~I- c~2)/2. This free energy expression predicts triple twinning (domain formation) in the low temperature phase and one can put Q2 = e5 = e6 = 0 for the domain I without loss of generality. The simplified free energy takes the form

a 2 b s C 4 1 Ml + Cr2\ 2 !. 0 2 F=Fo+2Ql+'3Ql+"4Ql+2\ 2 7 (el+ e2) +2 c 44 e4

+i cg6e~+ c~4e4e7 + Ae4Ql + Be7Ql + C(e1 + e2)Q~ + ••• (2)

For a free crystal aF/aQI = 0, two independent elastic instabilities result from the bilinear coupling terms between the order parameters and strains e4 and e7' In the high temperature phase, these elastic instabilities are represented as

and

a2 F 0 A2 0 T - Tl C44 =-;:;-::r = C44 - - = Cu

ae4 a T- To

a2 F 0 F 0 T- Tz C66=-;:;-::ra =C66--=C66 T T

e7 a - 0

with

and

B2 T2=To+~ •

ao C66

(3)

(4)

On the other hand, the elastic constant (c ll +CI2)/2 does not evidence an anomaly within the Landau theory; i. e.,

zF 0 0 ~_ a _CU+CI2 2 -a(el+e2)2- 2 (5)

In our previous Brillouin scattering experiments, the temperature dependence of cll was measured. As pOinted out in the Introduction, behavior of c ll can be decomposed into contributions from two independent sources as follows:

_ Cll + C12 Cll - C12 _~ c ll - 2 + 2 - 2 + CS6 .

Substituting the calculated effective elastic constants c66 and (c ll +c1z)/2 into Eq. (7), one finds

C~1 + Crg f.T - To)-/>+ 0 B2 c ll = 2 -A+ \ To +C66-~

(7)

_ 0 B2 A (T- To)-/>+ _ 0 T- T3 A (T- To~-/>+ - cll - - + - cll - + rp a ~ T-~ ~o (8)

in which T3 = To + (B2/ aOCrl). This is the same expression used to analyze the Cll anomaly in our previous report. 8 In Fig. 8, temperature dependence of C66 and (cu + C12)/2 are shown. It is clear from the figure that (c ll +CI2)/2 has the stronger anomaly. This elastic constant (c 11

+ CI2)/2, is derived by calculation from the measured Cll and C66 values in a manner similar to that shown in Eq. (7).

The half-width of the LA phonon parallel to the [100] direction has an anomalous increase near the transition temperature. 8 This anomaly is also due to anharmonic interaction with the order parameter fluctuations. Since the imaginary part of C66 is always smaller than the real part (i. e., the T A phonon is always well defined and under damped), the half-width anomaly must be primarily associated with the imaginary part of the elastic constant (cu + CI2)/2. This can be written31

r~l __ w_~' , 1m L 2 r2kBT~ C(q)C(q)

x [dteiwt(IQ1(Q,t)12IQl(q')12) , (9) -«>

in which wave vector and time dependence have been introduced into the coupling coefficient C and order parameter Q1 fluctuations. One can therefore conclude that the order parameter fluctuations play an important role in the dynamics of the phase transition in benzil.

Let us now turn to a discussion of the defect associ-ated central peak and relaxation time anomaly in benzil. In the experiment using the closed cycle cryostat, ex-ternal periodic stress is always applied to the sample. It should be emphasized here that: the correlation function could not be observed without the external pulse

12

.2

°80~----~~--~~----~----~I60

FIG. 8. Temperature dependence of the elastic constants (ell +cd/2 and css. The full straight lines are extrapolated from the high temperature limiting slopes. The vertical axis is in units of 1010 N/m2 •

J. Chem. Phys., Vol. 77, No.6, 15 September 1982


from the refrigerator even around the transition tem-perature; the correlation function only appeared for unannealed samples; and the well-defined correlation function was observed only in specific directions as-sociated with the Cu elastic constant already shown to evidence strong critical behavior. The first observation is consistent with the previous experimental results for correlation scattering in KDP,22 quartz,23 and Pb5GesOu.24 Although the correlation function is clear-ly induced by the external pulse, it is still possible that an intrinsic central peak for which a correlation function cannot be resolved by light scattering techniques does indeed exist in this system. Nonetheless, it is of con-siderable interest to understand how the external pulse induces the correlation function and by what mechanism the Rayleigh anomaly takes place around the transition temperature. Below a possible theoretical model de-scription which can qualitatively explain the observed facts is presented. In this discussion, we employed the theory of anelasticity as the observations are for a system under periodic external applied stress. The relation between these ideas and those of the usual cen-tral peak defect theory will be presented after this de-velopment.

Under the above conditions, a low frequency di~persion will be induced in the elastic constants by the forced motion of the local field around an imperfection. Therefore, the theory of elasticity in a perfect crystal cannot model the actual experimental conditions. In-stead, the theory of anelasticity32 should be employed to account for the induced dispersion. A stress applied perpendicular to the ac plane


r=y+aw!I(W~ 7e )

Near W = 0, one finds

(q2/p)(1+ iw7e) X(q,W)RaY1elgh'" w2(1+iw7 ) ,

a eff

with

(18)

(19)

The results for the susceptibilities near W = wa and (»=0, as given by Eqso (17) and (18), are very similar to those which can be derived from a microscopic model of the effect of defects or impurities on the static and dynamic response, developed by Halperin and Varma. 25 Details are given in the Appendixo The present macro-scopic description in terms of frequency dependent elastic constants seems, therefore, to be equivalent to the relaxing defect model of Halperin and Varma.

The light scattering structure factor in the high tem-perature limit35

S(q, w) = - (kB T/ w)Im X(q, w)

is then written as a sum of two parts:

S( ) - (kBTq2aw~7e1p) (1 2 2 )-1 q, W - 4 + W 7 ef! Wa

+ (W~ + aW~ _ W2)2 + r 2w2 (20)

Here, the first part gives the central peak response and the second term the LA-phonon peak. In Eq. (20), the effective relaxation time is given by Eq. (19)0 USing Eqs. (15) and (16), as well as the fit of previous mea-surements of c ll = cf1 to Eq. (15),

Cll = (1. 58 x 1010 N/m2)( T - 73.4)/( T- 72.6) ,

one finds

In a similar manner, one can calculate the integrate intensity of the central peak as

I (q2/p) aw~ _ aCll (T-72. 6)2

ex 4 - 'i":1l\2 • wa (Cll) T-73.4

(22)

The response of the annealed crystal also follows from Eq. (20) by taking the limit aw~ - aCll - O. It is seen that the central peak at W = 0, as given by Eq. (22), then disappears and only the Brillouin peak at W = wa and with width y remains. This also follows from the Hal-perin-Varma theory as outlined in the Appendix, if the defect concentration C vanishes. Also, the temperature dependent anomaly for 7eff disappears. For the un-annealed crystal, Eq. (20) predicts slight changes in the Brillouin frequencies and linewidth from the original values of Wa and y to v'w~+aw~ and r=y+aw!I(w: 7e ), respectively. As discussed above, however, no such changes are observed; these differences probably fall within the experimental uncertainty.

Numerical calculations using the above expressions, assuming that aCl1 and 7 e are not functions of tempera-ture, reproduce a weaker temperature dependence than the observed anomalies evidence. This is perhaps not surprising in view of the mean-field nature of our de-

scription. 25

To test the theoretical relations given by Eqso (19) and (22)

(23)

with cll constant around the tranSition temperature, we estimated the anomalous part of the effective relaxation time 7 eff by assuming 7 e possesses a monotonic tem-perature dependence through the phase transition as shown in Fig. 4 by the broken line. The calculated quantity (7 eff - 7 e)2 shows a sharp peak around the tran-sition temperature and qualitatively reproduces the in-tensity anomalies observed in Brillouin and correlation scattering which are shown in Figso 4 and 6.

The above result, that the intensity and square of the anomalous part of the relaxation time are proportional to one another, suggests that the observed central peak and correlation Should be extrinsic effects generated by the motion of annealable defects in response to the external applied stress from the refrigerator. This clearly separates the present observations from the intrinsic anomalies found for KDP, KTS, and Pb5Ge30 l1 • Without applied stress, no correlation function (relaxa-tion process) could be observed: the defects will re-cover their intrinsic dynamiCS, relaxational or frozen, which the present light scattering experiments could not resolve, and the complex elastic constant becomes the real elastic constant. In this case, the theory pre-dicts no central peak response and only the usual soft mode behavior is expected.

Of course, none of the above ideas or discussion pre-cludes or denies in any wayan intrinsic central peak behavior like that observed in KDP, KTS, etc., for benzil even if its origin is related to defect dynamiCS. In these reported experiments, however, no such in-trinSic central peak dynamicS could be observed.

The defect relaxation observed seems to occur only in response to the external applied stress from the re-frigerator. One can introduce two defect states to de-scribe the behavior of the defects following the applied stress: a relaxed or nonstressed state and a stressed or unrelaxed state. Transitions are caused between these two states by the external stress pulse only. Re-call that the stress pulse can be thought of as a square wave with a rise time of tens of jlS and an on time of roughly 100 ms. This conception of the present situa-tion is completely analogous to the introduction of a spontaneous hopping of defects between two possible states in the crystal. In this sense, the externally in-duced central peak phenomenon can be seen as an ex-ample of the "relaxing defect cell" model in the theory of Halperin and Varma. 25 Once such a binary state sys-tem is introduced into the crystal, by whatever intrinsic (spontaneous) or extrinsic (external force induced) mechanism, a central peak response function follows for the system. This added defect degree of freedom is the essential feature of a defect induced central peak phenomenon. This is the major reason that the present theoretical results are parallel to the more usual25 non-stress induced defect related central peak response function.



For the scattering geometry shown in Fig. 7 (sample II), the applied stress O's in the ab plane induces a combina-tion of strains,

O's = CISel + czsez + csses • In this case, Cll has no low frequency dispersion or

response. Moreover, the elastic constants appearing in the above expressions are not included in the elastic constants related to a-axis phonons.

Thus, the above developed theory gives a qualitative explanation of the observed defect related central peak and relaxation time found in this system. For a quanti-tative discussion of these phonomena, more precise measurements of the elastic anomalies, relaxation times, and Rayleigh intenSity are required.

V. CONCLUSION

The a-axis polarized central peak observed in our previous Brillouin scattering experiments has been ex-tensively studied by Simultaneous measurement of Bril-louin scattering and correlation spectroscopy. Our new findings are summarized as follows:

1. Correlation spectroscopy reveals a slow relaxa-tion process in as grown samples only when an external stress is applied to the sample. On the other hand, well-annealed samples do not show the relaxation pro-cess under the same conditions.

2. The induced relaxation process can be observed in polarized scattered light and depends on the direction of the applied stress and on the observation direction.

3. The relaxation time and Rayleigh intenSity show anomalies around the transition temperature, as re-vealed by the previous Brillouin experiments.

4. Brillouin Shifts and elastic constants are found to be independent of the thermal treatment of samples to within the experimental uncertainty (-2% for ~VB' -4% for coff)'

These results suggest a possibility of annealable local stress fields due to defects or impurities in benzil gen-erated during the processes of crystal grOwing and shaping. Some of these fields can be forced to deform and recover their shapes in response to an external applied stress pulse.

To formulate this idea more quantitatively, a complex elastic constant c:'t(w) is introduced through the theory of anelasticity. A dynamical susceptibility for LA phonons propagating along the [100J direction is dis-cussed. This susceptibility can be approximately de-composed into a Rayleigh component which shows in-tenSity divergence and critical slowing down of the re-laxation time and a component which shows a pair of LA phonon responses. Within the theory, the central peak phenomenon can be related to an anomaly in the elastic constant cll . The c 11 anomaly arises from two separate contributions: a weak C66 anomaly and a strong anhar-monic coupling between the strain (el + ez)/2 and fluctu-ations of the order parameters over wide temperature range (T :STc +50 K).

Although the theory qualitatively explains the ob-

served characteristic features, numerical calculations employing the previous results for the cll anomaly re-veal weaker divergences than are actually observed. The observed temperature dependence of the relaxation time is stronger than that estimated from the elastic anomaly. To improve the fitting, higher order inter-actions between the order parameters, including both the zone center and zone boundary modes, and various strains seem to be necessary.

The observed intensity anomaly in as grown crystals can be understood as an external applied stress induced effect. This phenomenon may also be modeled by the usual expression for the central peak response function which consists of a soft mode response with a relaxing self energy. Such an approach leadS to the same ex-preSSions as those generated by the theory of anelastic-ity presented in this work. In this sense, there exists a strong relationship between the two approaches to central peak behavior. The central peak response function also predicts a relation Icr.(~T)z between the intenSity divergence and the relaxation time anomaly just as observed in these experiments.

Finally, a number of other samples have been studied and this phenomenon of a central peak in unannealed samples is not unique to benzil. Chloranil and urea show similar effects while NH4CI and triazine do not. Such observations are, of course, related to intrinsic (anisotropic) dynamiCS, defect concentration, and de-fect nature. These additional observations will be pub-lished in' the near future.

After submission of this manuscript, a paper appeared dealing. with x-ray diffuse scattering for benzil: H. Terauchi, T. KOjima, K. Sakaue, F. Tajiri, and H. Maeda, J. Chem. Phys. 76, 612 (1982). Theseworkers report critical diffuse scattering for both the zone bound-ary (M pOint) and zone center instabilities. These re-sults support, in general, Toledano's predictions and our previous analysis of the C11 anomaly for benzil.

APPENDIX: COMPARISON WITH RELAXING DEFECT MODEL OF HALPERIN AND VARMA

The mean-field equations as developed by Halperin and VarmaZ5 which describe the response of a relaxing defect are, in their notation,

~)Oa/!-Ja/!(q)Xa(w)JYa(q,w)=Xa(w)ha(q,w). (AI) /!

Here, Ya(q, w) and ha(q, w) are the Fourier transforms of the particle displacement and the local external field, respectively. The indices a, (3 run over nand d to in-dicate normal and defect cells, respectively. For a concentration of defects c,

Yn=(l-c)


present work, we instead assume

J",,(q) =J/fd(q) =0,

J,.,(q) = Jd,,(q) =J(q) • (A3) Here, J(q) represents the coupling between a particular (normal) phonon mode, with the susceptibility deter-mined from

x;,1 = w~ - w2 + iyw (A4) and a diffusive mode, to represent the defect motion, with a susceptibility given by

x.;1 = kB T(l + iWT e) • (A5) Although it is apparent from the present work that the coupling J(q) between the (normal) phonon modes and the defect motion depends strongly on phonon polariza-tion and propagation direction, the details of this de-pendence remain to be elucidated. Releasing condition (A3) on the diagonal coupling terms would only result in a renormalization of the phonon susceptibilities and frequencies.

The various coupled susceptibilities are then calcu-lated from Eq. (A1). For example, the phonon-phonon susceptibility, defined by

(A6)

near W =Wa, is given by the same expression as Eq. (17) with

(A7)

Near W = 0, one finds

(A8)

with ~w~ given by Eq. (A7) and

Te!f~ ~+ ~w1:)Te' (A9) This expression for the Rayleigh susceptibility becomes identical to Eq. (18) for ~w!« w~.

The other contributions to the susceptibility of the coupled system, i. e., x"d, >i.!", and >i.!d' have the same denominator as x"" and, therefore, can be expected to make qualitatively similar contributions.

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Critical behavior in annealed and unannealed crystals of ...Benzil (C6HsCOh is a molecular crystal which has a crystallographic phase transition at -83.5 K. Since the discovery of